Radon-Nikodym Derivatives of Gaussian Measures Author(s): L. A. Shepp Source: The Annals of Mathematical Statistics, Vol. 37, No. 2 (Apr., 1966), pp. 321-354 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2238606 Accessed: 19/11/2010 14:13

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http://www.jstor.org RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES

BY L. A. SHEPP Bell TelephoneLaboratories, Inc., MurrayHill, New Jersey I. SUMMARY We give simple necessary and sufficient conditions on the mean and covari- ance for a Gaussian measure to be equivalent to Wiener measure. This was formerly an unsolved problem [26]. Another unsolved problem is to obtain the Radon-Nikodym derivative d,/dv where u and v are equivalent Gaussian measures [28]. We solve this problem for many cases of , and v, by writing du/dv in terms of Fredholm determinants and resolvents. The problem is thereby reduced to the calculation of these classical quantities, and explicit formulas can often be given. Our method uses Wiener measure ,w as a catalyst; that is, we compute deriva- tives with respect to ,uwand then use the chain rule: d1./dv = (d,4/dsw)/(dv/dg w). Wiener measure is singled out because. it has a simple distinctive property-the Wiener process has a random Fourier-type expansion in the integrals of any complete orthonormal system. We show that any process equivalent to the Wiener process W can be realized by a linear transformation of W. This transformation necessarily involves sto- chastic integration and generalizes earlier nonstochastic transformations studied by Segal [21] and others [4], [27]. New variants of the Wiener process are introduced, both conditioned Wiener processes and free n-fold integrated Wiener processes. We give necessary and sufficient conditions for a Gaussian process to be equivalent to any one of the variants and also give the corresponding Radon-Nikodym (R-N) derivative. Last, some novel uses of R-N derivatives are given. We calculate explicitly: (i) the probability that .W cross a slanted line in a finite time, (ii) the first pas- sage probability for the process W(t + 1) - W(t), and (iii) a class of function space integrals. Using (iii) we prove a zero-one law for convergence of certain integrals on Wiener paths.

TABLE OF CONTENTS Page I. SummARY ...... 321 II. INTRODUCTION ..322 1. A necessary and sufficientcondition that ,u .w 322 2. The R-N derivative d/ds. .323 3. A general representation for W .324 4. Simultaneous representation of , anddw .326 5. Calculating dc/dv..327 Received 19 November 1965. 321 322 L. A. SHEPP

6. Conditioned Wiener processes ...... 329 7. Stochastic linear transformations of W ...... 330 8. A discussion of previous work ...... , ...... 332 9. Double Wiener integrals ...... 332 III. PROOFS OF THE MAIN RESULTS ...... 334 10. Proof of Theorem 1; Sufficiency ...... 334 11. Proof of Theorem 1; Necessity ...... 336 12. Proof of Theorems 2 and 3 ...... 338 13. Proof of Theorem 7 ...... 339 14. Proof of Theorem 8 ...... 341 IV. EXAMPLES OF R-N DERIVATIVES ...... 344 15. Scale changes of W ...... 344 16. The linear covariance ...... 346 V. SOME APPLICATIONS OF R-N DERIVATIVES ...... 348 17. Calculating probabilities ...... 348 18. Evaluating some special integrals ...... 350 19. A dichotomy theorem for Wiener paths ...... 352 II. INTRODUCTION All measures considered in this paper are Gaussian and are considered to be defined on the space of continuous functions X = X(t), 0 < t < T with T < 00. Such a measure Auis determined by its mean m and covariance (cov) R m(t) = f X(t) d,u(X),

R(s, t) = f (X(s) - m(s) ) (Xi(t) - m(t) ) d,u(X).

Wiener measure Auwis the measure with mean zero and cov = min (s, t). Two measures are equivalent (denoted -) when they have the same sets of measure zero.

1. A necessary and sufficient condition that w_g W. We denote by L2 and L2 the space of square-integrable functions on [0, T] and [0, T] X [0, T] respec- tively; two functions are considered equal if they coincide almost everywhere. Suppose Auis a measure with mean m and cov R. THEOREM 1. , - AuWif and only if there exists a kernel K E L2for which (1.1) R(s, t) = min (s, t) - J' ' K(u, v) du dv and (1.2) 1 e -(K) and a function k E L2 for which (1.3) m(t) = f'k(u) du. The kernel K is unique and symmetricand is given by K(s, t) = -(d/ds) (d/dt). R(s, t) for almost every (s, t). The function k is unique and is given by k(t) = m'(t) for almost every t. RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 323

As usual, o-(K) = {X: K

Since K is symmetric and in L2the eigenvalues X1, X2, * are real and E X12 < . We shall show in Section 11 that if R is given by (1.1) with K E L2, then R is nonnegative-definite if and only if Xj < 1 for all j. The condition (1.2) therefore says that Xj < 1 for all j and should thus be interpreted as a statement of strict positive-definiteness for R. Note that Xi may assume negative values. The condition (1.1) has a simpler restatement in case Ri(s, t) = (a/as)R(s, t) is continuous for s # t. In this case (1.1) becomes (see Section 11 for the proof)

(1.4) R1(s, s+) - R1(s, s-) 1, 0 < s < T. This means that R must have a fixed discontinuity of unit size in its derivative along s = t, the same discontinuity that min (s, t) has in its derivative. A theorem of G. Baxter [2] implies the necessity of (1.4). We note that R (O, *) 0 because X(O) = 0 under Aw . 2. The R-N derivative d,Id,w. Whenever ,u ,uw the R-N derivative d1z1d1uw exists. We will show that d,u/d,uw can be written in terms of the Fredholm de- terminant and resolvent of the unique kernel K appearing in (1.1). When E lXjl < oc, K is said to be of trace class and the Fredholm determinant is (2.1) d(X) = fJj (1 - XX,). For the general K, E lXjl may not exist. The modified, or Carleman-Fredholm, determinant of K is

(2.2) 6(X) = Ij, (1 -Xj)ei which converges for all X because E Xj2 < oo. For each value of X for which X-19 a(K) there is a unique kernel H, e L2 called the Fredholm resolvent of K at X. The resolvent equation

(2.3) Hx - K = XHxK = XKHx determines HA uniquely. We denote H1 by H for simplicity and note that H is defined because 1 z o(K) by (1.2). The kernel H is symmetric and is continuous when K is continuous. There are known expansions of d, a and H in powers of X, cf. [7], pp. 1081-1086. Let ,ube a measure with mean m and cov R for which ,u,u'w . Let K be given by (1.1): K(s, t) = -(d/ds) (O/t)R(s, t) for almost every s and t. THEOREM 2. If K is continuous and of trace class then d,u/d,uwis given by

(2.4) djA/djAw(X+ m) = [d(1)]-2

*exp [-2 rO rO H(s, t) dX(s) dX(t) + rO k(u) dX(u)+ 2 J' k2(u)du] Here k(t) = m'(t). When m = 0, (2.4) simplifies. The integral fT k(u) dX(u) 324 L. A. SHEPP

is the Wiener integral evaluated at the point X. It exists because k e L2. The

integral I(X) = rO rO H(s, t) dX(s) dX(t) is the double Wiener integral evaluated at X and is a (non-Gaussian) random variable with mean value JO H(s, s) ds. In our case the integral I can be defined because H is continuous. If H is also of bounded variation, we may integrate by parts to obtain an ordinary integral. Both H and the constant d( l) may sometimes be obtainable in closed form even when o(K) is not so obtainable, as we shall see in Section 15. In order to give a formula for d,u/d,uwvalid for all K, the notion of a double Wiener integral must be extended slightly. We will use the centered double Wiener integral denoted

J(X) = f~'0cJ H(s, t) dX(s) dX(t). J is introduced in Section 9. When the mean of I exists, J is the ordinary double Wiener integral I minus its mean. By this simple trick of subtracting off the mean, J can be defined for all H e L2. K. Ito [10] was the first to consider the centered multiple Wiener integral. He obtained it in an equivalent way, by ignoring the values of H on the diagonal. THEOREM 3. If A ,u ow then (2.5) (dM/dMw)(X + m) = (a(1) exp tr (HK)'

*exp [-4'J(X) + f ' k(u) dX(u) + TJ'1c(u) du]. The trace of a product always exists and we have

(2.6) tr (HK) = fT fT H(s, t)K(s, t) ds dt. When the hypothesis of Theorem 2 holds, (2.4) and (2.5) agree. Of course (2.4) is simpler. (2.5) has the advantage of being valid in general.

3. A representation for W. Let h, '12, *** be a sequence of independent standard normal variables (mean zero and variance one). Let (Pl, 2P2, ...* be an arbitrary complete orthonormal sequence in L2[0, T] and set (3.1) f';(t) = rosj(u)du, j = 1,2, THEOREM 4. For each t, 0 < t < T, the series (3.2) J=1 X()=Wt convergesalmost surely. The sum is the Wiener process on [0, T]. This result appears to be new except for two special cases due to Wiener and L6vy. To prove the theorem we observe (3.3) )j(t) = ((Pj, 1t where lt is the indicatorof the interval [0, t] and (f, g) = fT f(u)g(u) du. The completenessimplies

(3.4) Ej ('pj , 18)(vp , 1) = (ls , 1t) = min (s, t) RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 325

and (3.2) follows immediately from the 3-series theorem since the sum is Gaussian and has the required mean and covariance. Wiener himself studied the special case of Fourier series:

(3.5) 4b,(t) = 21(sin (j-1)-rt)l(j-2), 0- t < 1. In this case 4>as well as so are orthogonal and this property characterizes the Fourier case. In the Fourier case, (3.2) converges uniformly in t with probability one. However, for some other choices of the sequence (pthe convergence of (3.2) is even better. In fact, there does not seem to be any particular advantage to (3.5). To prove this, let us take soto be the Haar sequence. Using double indices for convenience, set

sp41 = 1, Son,j(t) = 2nl2, 2nt (j,j + 2

(3.6) = -2nI2, 2nte( + 2 j+ 1)2

= 0, otherwise

forj = 0, 1, ... , 2n - 1; n = 0, 1, 2, *. . The s's are complete and orthonormal. The representation (3.2) in this case is due to L6vy, cf. [11], p. 19, and takes the form

(3.7) W(t) = rt + En=o Z 0 ?n,jAn,j(t) where An,j(t) = 1n,j(t) = 0 'Pn,jis an isosceles triangle with base 2- centered at (j + :)2-n, and height, 2 - 2-n2. This shows that the Wiener process is a random sum of triangles. In Wiener's case, the series (3.2) fails to converge absolutely. By contrast (3.7) converges absolutely uniformly with probability one, as was pointed out by Z. Ciesielski. We follow [11], p. 19. Let fn(t) = Zj-o n We have (3.8) Ifn(t)I< 2-nl2 max, 1jinlj 0 < t < 1

because j An,j(t) < 2-n12 uniformly in t. Now Mn= maxj |?n,j| is the maximum of 2n independent standard normal variables and so

(3.9) P{Mn < an) = (I(an) -_(_an))2n. Choosing an = 2n' it is easy to check that

(3.10) Zn==PIMn > an} < 00. By the Borel-Cantelli lemma we see that Mn < aneventually and so, a.s.

(3.11) 1 2-'n2Mn < 00. It follows that (3.7) converges absolutely uniformly. H. P. McKean has informed me that by using a recent theorem of J. Delporte 326 L. A. SHEPP

[30], p. 201, Corollary 6.4B, it can be shown that (3.2) converges uniformly a.s. We will not use this strong type of convergence and so we omit the proof.' 4. Simultaneous representation of , and ,uwin terms of independent random variables. Let ,u be a measure with mean m and cov R satisfying (1.1)-(1.3). Let1, (p2, * * * be the eigenfunctions of K; these are orthonormal and complete. The representation (3.2) gives (4.1) W(t) = Zj14j(t), 0 < t ? T.

We shall define a Gaussian process Y on the same space as 1, X72, ... with mean m and cov R. Write

(4.2) k = mr= Ejkjrj, kj= (k, pj), and define (4.3) Y(t) = ,j (7j(l - Xi) + ki)'Ij(t). Y is clearly Gaussian and has mean

(4.4) EY(t) = Zjkj((pJ, 1t) = (k, 1t) = m(t) and covariance

(4.5) E (1 - Xj)4j(s)>1(t) = min (s, t) - fBfJt K(u, v) du dv = R(s, t). In (4.5) we have used the L2 expansion of K (4.6) K(s, t) = E Xjcpj(s)soj(t). We have proved the following theorem. THEOREM5. The processes (4.1) and (4.3) give a simultaneous representation of W and Y in termsof sums of independentvariables. It is now possible to give a formal expression for dl/dMtw(X). We expand a path X as (4.7) X(t) = ,jX4(t), Xi = fJo ,j(t) dX(t) where Xi = Xj(X) is the Wiener integral evaluated at X. du/d,w(X) is the relative likelihood of X under , and ,w . Under ,Mw,Xj = qj , independent random variables. Under A, Xj = j(l - Xj) + k1 also independent. Because of inde- pendence, the probabilities multiply and we get (4.8) (dI/dAw)(X) = IY=1(1- *exp [-l(Xj - kj)2/(1 - Xi)]/exp [-2 X,2]. The product (4.8) always converges and represents du/d,w. A rigorous proof is given in Section 10. In Section 11 we show how (4.8) reduces to (2.4). The reduction to (2.4) is important because (2.4) is in terms of classical quantities and, in addition, does not explicitly involve the eigenvalues or eigenvectors, which are usually difficult to find. IJohn Walsh has found a shorter proof, based on an abstract martingale convergence theorem. RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 327

5. Calculating dM/dv. Suppose that, 1t and v are measures, both equivalent to Aw . Then ,u - v and by the chain rule (5.1) (dtl/dv) (X) = (d/2/dAw) (X)/(dv/dtiw) (X). Applying (2.5) or (2.4) to obtain d,l/dMwand dv/dAwwe get an explicit formula for dM/dv,but only in the special case when A -, iw and v - Aw . To get more general results we will study certain variants of W.

Letforn=0,1,2, 12, I (5.2) Wn(t) = [(t - u)n/n!] dW(u), 0 ? t < T denote the n-fold integrated Wiener process. We have

(5.3) Wo(t) =W(t), = du, n = 1, 2, *** VWn(t) foWn-.(u) The processes Wn satisfy W.(j)(0) =0, 1j = 0, 1, ... , n. Suppose ,u is a measure with mean m and cov R whose sample paths Y are n-times differentiable a.s. Let y(n) denote the nth derivative of Y. The process y(n) is Gaussian with mean m(n) (t) and covariance DlnD2nR(s, t) = (an/asn)(an/atn)R(s, t).

Let i(n) be the measure induced by y(n); A(n) has the same mean and covariance as y(n) THEOREM 6. Suppose jt -- ,wn . Then ,(n) r w and

(5.4) (dAIldALw')(X) = (dAU(n)1dAXW)(X(n) ).

The mapping X - X(n is 1-1 on the set of n-times differentiable functions X for which X(j)(0) = 0, j = 0, 1, ..* , n and standard arguments [9], pp. 163-164, give (5.4). The righthand side of (5.4) is given by (2.5). Using (5.4), we can obtain dl./dv explicitly whenever - v '' Awnfor some n. This generalizes (5.1) and, further, one may drop the assumption that n in (5.2) - is an integer. However, the condition , . v 'uwn is still too restrictive. Ex- cluded are stationary measures ,u and v because their sample paths do not vanish, at zero. In order to remedy this lack we must unpin the process Wn at zero. Let Wn be the free Wiener process (5.5) Wn(t) = 7=o0jt'/j! + Wn(t), 0 < t < T, where to, ***, tn are independent, standard normal variables. Suppose Y is a process for which Y - Wn . The paths Y(t) are then exactly n-times differentiable, and the derivatives of Y at t = 0 are nonzero random variables. The class of processes Y ,- W. includes many processes of interest. We will see that stationary processes with rational spectral density are included as a special case. In the latter case, d,i/dv has already been found by Gelfand and Yaglom in a different way [28]. We shall give the conditions on m and, R so that Y -Wn . The condition on m is that m(n+1)e L2 or 328 L. A. SHEPP I (5.6) m(t) = Xs=o [mI1(O)/j!]t' + f [(t - U)n/n1]k(U) dU, where k c L2. In order to find the conditions on R, we first obtain a certain decomposition of an n-times differentiable covariance. The remainder of Section 5 will be needed only for Section 13 et seq. An n-times differentiable process Y is called nondegenerateat zero when the random variables Y(O) , Y'(O) , ... , Y(n (O) are linearly independent. THEOREM 7. The covarianceR of an n-times differentiableprocess nondegenerate at zero may be written uniquely as (5.7) R(s, t) = Z"=oAi(s)Ai(t) + R*(s, t) where (i) R* is a covariance, (5.8) (ii) DliD2iR*(0, 0) = 0, i = 0, 1 n, (iii) Aj(t)(0) = 0, i < j; A i $(0) > 0 i = 0 ,n. Condition (ii) means that if Z is a process with cov R* then (5.9) Z(,'(0) = 0, j = 0,1, ... , n.

We will have Z -'Wn . The decomposition (5.7) is designed to reduce the prob- lem to Theorem 6. There are elegant formulas for A,oX , An closely related to the decomposi- tion formulas in Gauss's elimination method. To obtain them suppose R is a covariance that is n-times differentiable in each argument. Define (5.10) Rij(s, t) = DliD23R(s, t), Rij = Rij(O, 0). Let a-, = 1 and for i > 0 let Roo ... Roi (5.11) ai= . . . Rio ... Rii

If R(s, t) = EY(s) Y(t) where Y is n-times differentiable then as is the Gram- mian of Y(0), **, y(i)(o) and ai is strictly positive when Y is nondegenerate at zero. Whenever ai > 0, i = 0, * , n the functions Ao, n, A4 in (5.7) are unique and are given by Roo ... Roim-iRoo(O) t) (5.12) A;(t) = ( at-i)iR1i-1 R0(0,t) , Rio * *Ri;m-i Rio(O)t) i = 0, 1, ..., n. In particular, Ao(t) = R(O, t)/(R(O, 0))'. Note that Ai is independent of n for n > i. Define for i = 0, ,n, j = *, , n, Roo * Roi-i Roj (5.13) Aji = Ai(j)(0) = (atiai-1)i.. .. Rio ... Rii-1 Rij RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 329

The (n + 1) X (n + 1) matrix A = Aij is lower semidiagonal and has an in- verse. The inverse matrix is denoted by C and can be written explicitly (13.6). The next theorem gives the conditions for a measure ,u to be equivalent to ;tw. as well as a formula for d,u/d;twnwhenever it exists. It is clear that whenever ,u -,pw the path functions Y must be n-times differentiable and nondegenerate at zero. In this case the covariance R of ,u satisfies the hypothesis of Theorem 7, and R* and Ao, * **, A. are uniquely defined. THEOREM 8. Let ,ube a measure with mean m and cov R. ,u-- ,Awnif and only if: m satisfies (5.6), R has a unique decomposition (5.7), and

(5.14) DinD2nR*(s, t) = min (s, t) - fo rO' K (u, v) du dv for a (unique, symmetric) kernel K e L2 with 1 z a (K) and (5.15) A.(n)(t) = f ai(u) du + Ai(n'(O)

for (unique) aO , a. in L2. When ,- gw. the R-N derivative is given by (5.16) provided that K satisfies the conditions of Theorem2:

(dA/dMwj)(X + m) = [d(l)aof?exp [-2jo dX0ff(s)ftdX04)(t) H(s, t)

+ rO~'k(u) dX )(u) + X(k, k)

(5.16) - 4E7.=o (e2 - (X(j)(0) + m(j)(0))2)

- E = Eo =oeeje ((I + H)ai, aj)

+ i$'=o eiJoT ((I + H)ai(t)) dX(n)(t)]. Here I is the identity on L2, H is the resolventof K at X = 1, d(*) is the determinant of K, and

(5.17) En oci (x(i)(0) - m(i)(0)), j = 0, 1, ... , n, where C = A-', the inverse matrix of A in (5.13). For general K we must replace the double stochastic integral in (5.16) by the centered integral and replace d(1) by 8(1) exp tr (HK). This modification is completelyanalogous to that of Theorem3. (5.16) is cumbersome. Its importance lies in its generality rather than in its simplicity; in many cases dM/d.w. can be obtained more simply by other means. 6. Conditioned Wiener processes. We next consider sub-Wiener processes, obtained from W by linear conditioning. The Wiener integral

(6.1) X =P (y)=o (t) dW(t) is defined for VI L2; v is normal with mean zero and variance (t, #) = T f f2(U) du. Let V be any subspace of L2 = L2[0, T] and let v be the (Gaussian) measure obtained by conditioning Auwso that v(+I') = 0 for , e V. Then 7Avhas mean zero and cov (6.2) Rv(s, t) = min (s, t) - *1(s)Ij(t) 330 L. A. SHEPP where

(6.3) *j'(t) - ,f,jf (u) du and #j, 2 * is any orthonormal basis for V, contained in V. We may realize a process with measure uV as follows: (6.4) WV(t) = W(t) - Zj*j(t)Wj,, Wi = r7(0j)

It is easy to check that WV has mean zero and cov R . Alternatively, let P1, (o2, ... be an orthonormal basis of the orthogonal complement of V in L2. Of course, - - - Pl, ) 2 * * are not complete. Let '1l, 'l2, be independent standard normal variables and define 1Djas in (3.1). Then another realization of WV is (6.5) WV(t) = ? jl)j(t). The interest in (6.4) is that it is a realization on the same space as the original process W. As an example, take V to be the space generated by + = = 1. Then with T = 1, J(t) = t and (6.2) is

(6.6) Rv(s, t) = min (s, t) - st. Now W1 = n(0) = f dW(t) = W(1) and (6.4) gives (6.7) WV(t) = W(t) - tW(l), 0 < t < 1. The process (6.7) is called the pinned Wiener process [5]. The processes (6.4) are mutually singular for different subspaces V, and V2 . Indeed, if l ? V1 but VIe V2 then n(#/) = 0 for ,iv1 but is normal with nonzero v' variance for AV2 so that li IAV2. What is the condition on u. so that A AV? The answer is given by the next theorem. THEOREM9. ,u Vu if and only if

(6.8) R(s, t) = Rv(s, t) - fI f K(u, v) du dv where K c L2 and in addition (6.9) ea (K) and K*= 0 for IeV. The mean must satisfy (1.3) and in addition

(6.10) (k, A) = 0, ?V. In case (6.8)-(6.10) hold the R-N derivativeis given by (2.5) whereK is the unique kernel satisfying (6.8). When-K satisfies the conditions of Theorem 2, (2.4) is also valid. 7. Stochastic linear transformations of W. Interesting classes of processes can be obtained by various linear transformations of W. Segal and others con- sider some nonstochastic transformations and with them realize some processes equivalent to W [21], p. 464. By means of a transformation depending on a stochastic integral we will realize any process equivalent to W. RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 331

Suppose M e L2. The stochastic integral (7.1) Z(s) = f M(s, u) dW(u), 0 < s < T can be defined for each s in such a way that Z is a.s. measurable [6], p. 430. As- suming Z so defined, we set (7.2) Y(t) = W(t) Z(s) ds and call Y the affine transformationof W with kernel M. Of course when M is of bounded variation in the second argument, we may write Y as an ordinary non- stochastic transformation by integrating by parts. The process Y is Gaussian and has mean zero and covariance (7.3) R(s, t) = rmn (s, t) - f'f ' K(u, v) du dv where (7.4) K = M + M*-MM*. As usual, M*(u, v) = M(v, u) and, of course, MM*(u, v) = JYM(u, y)M(v, y) dy.

Let I denote the identity. Then by (7.4), 1 - K = (I - M)(I- M*). Since a(M) = a(M*), (7.5) 1 r (K) X 1 e (M).

By Theorem 1 we obtain: Y --. W if and only if (7.6) 1 Ka(M). Suppose R is any covariance satisfying (1.1) and (1.2). It is simple to show that there is an M satisfying (7.4) and (7.6). Indeed, the kernel K of R has the Mercer expansion in L2, (7.7) K(s, t) = i Xjspj(s)pj(t) where Xj < 1 for all j. We may take M to be (7.8) M(s, t) = , [1 d=(1 - XjA'Pj(s)Vj(t). When all but a finite number of the signs in (7.8) are negative we have M e L2 and it is easy to see that M = M* and (7.4) and (7.6) hold. We have proved the following theorem. Let ui be Gaussian with mean m and cov R. THEOREM 10. Suppose ,I -,uw. There is an M for which the process Y + m, where Y is given by (7.2), is a realization of ,u. In other words, the measure 11Y+m induced by Y + m satisfies

(7.9) ILY+m = j. M is not unique. There are additional conditions one could put on M in order to make it unique in Theorem 8. However, none seems natural. 332 L. A. SHEPP

When M is Volterra (7.10) M(s, t) = 0 for s _ t the process Y is causal: Y(t) depends only on values W(-r) for Xr _ t. But we could not solve the problem of the existence and uniqueness of kernels M of Volterra type in (7.4). 8. A discussion of previous work. I. Segal considered a nonstochastic trans- formation S of W [20], p. 22; [21], p. 464. He defined (8.1) S(t) = W(t) + f ' N(t, u)W(u) du, 0 < t < T. N is assumed continuous and Nt e L2. The transformation (8.1) generalizes an earlier one of Cameron and Martin [8]. It is easily seen that (8.1) is a special case of (7.2). Segal asserts that S - W under the stated conditions on N. How- ever, note that a spectral condition analogous to (7.6) is needed. Even if this correction is made, (8.1) is not "best possible" as Segal claims. The stochastic transformation (7.2) is better because it gives the most general transformation equivalent to W (Theorem 10). In the case IA = $Aw+m,X a translate of ,uw, the condition (1.3) as well as the formula for d,u/d,uw was known and is due to Cameron and Martin [6], and in the general case to Segal [21], p. 462. D. E. Varberg [26] gave a formula for dA/dAwwhen R admits a certain factoriza- tion. His formula is based on a transformation of Woodward [27], which is simriilarto (8.1). Varberg requires many complicated additional assumptions. These complications arise because: (a) his approach is based on a transforma- tion which is not general enough and (b) d,u/d,uwitself depends on the transforma- tion only through the covariance of u-many transformations give the same covariance as is shown by the manifold nonuniqueness of M in Theorem 10. For this reason it is better to work with the covariance directly. It has been brought to my attention by R. H. Cameron that D. E. Varberg in an unpublished manuscript has independently obtained an equivalent form of the sufficiency half of Theorem 1. We should mention that a condition similar to (1.1) and (5.14) appears in work of Yu. A. Rozanov [19], p. 455. We acknowledge with pleasure an informative private lecture given by J. Feldman and L. Gross and many profitable discussions with S. P. Lloyd.2 9. Double Wiener integrals. We will now define the centered double Wiener integral " (9.1) J~~~(X)-f cf H(s, t) dX(s) dX(t) for H E L2 and for almost every function X. A simple function H has the representation (9.2) H(s, t) = X >I ajkxjk(s,t)

3Note added in proof: A. M. Yaglom communicated that I. M. Golosoy recently obtained results, announced in Dokl. Akad. Nayk CCCP 166 (1966) 263-266, which should be com- pared with ours. He obtains, among other things, an equivalent version of our Theorem 1 RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 333

where

(9.3) Xjk(82 t) = 1 (82 t) e (t_-1 tj) X (tk-1 X tk) = 0 otherwise for some partition, 0 = to < ti < ... < t. = T. When H is simple define

(9.4) I(X) = EZr 1 ajk(X(tj) - X(tj..l))(X(tk) -X(tk-l)) and (9.5) J(X) = I(X) - foH(s, s) ds. It is importantand also easy to check thatJ = JH does not dependon the parti- tion used to define it. J is not normally distributed, but has mean zero and variance (9.6) f J2 d,w = 2f foH2(s, t) ds dt. Now suppose that H is any element of L2 and that Hn is a sequenceof simple functionsfor whichHn -+ H in L2.By (9.6),JH. is a Cauchysequence in L2(,4W). We define JH = lim JH . It is easy to check that JH does not depend on the sequenceHn used to define it. We call J the centeredintegral because (9.7) EwJ = fJdw = O. The c betweenthe integral signs in (9.1) calls attention to (9.7). For later use we point out that for H(s, t) -p(s)'p(t),= a degeneratekernel, we get

(9.8) efcfo p(s)vp(t) dX(s) dX(t) = q'(4p) - ((p, (p) where q(qp)is the single Wienerintegral (9.9) j((p) Jof p(t) dX(t). The (uncentered)double Wiener integral

(9.10) I(X) = J'oTfJ H(s, t) dX(s) dX(t) is now easy to define.We defineI = IHfor continuousH by (9.11) I(X) = J(X) + f ' H(s, s) ds. There is an important formula for I obtained by two integrationsby parts when H is of boundedvariation. We say H is of boundedvariation when (9.12) Var (H) = sup Z Z jH(tj,H tk) - H(tj_i Xtk) - H(tj Xtk-l)

+ H(tj_l X tk-1) is finite, the sup being taken over all partitions.In this case we have (9.13) I(X) = H(T, T)X2(T) - X(T)fJ X(s) d8H(s, T)

-X(T)f o X(t) djI(T, t) + rO rO X(s)X(t) d, dtH(s, t). by a rather different method. He then applies the theorem to obtain conditions for equi- valence and singularity for an arbitrary Gaussian measure and a Gauss-Markov measure. 334 L. A. SHEPP

One proves (9.13) first for simple functions and then for general H of bounded variation by passage to the limit. The advantage of (9.13) is that it does not involve stochastic integration and so is defined for every continuous function X.

III. PROOFS OF THE MAAINRESULTS We first obtained Theorems 1-3 and 9 heuristically, using a direct evaluation of d,u/d,uwas a limit of finite-dimensional densities,

(d,u/dAuw)(X) = lim [p((xi, , xn)/Ppw(Xi,w . , x,)], xi = X(ti), where =() (27r)-f'2jRj- exp (R-'(X - in), (x - )) The limit is taken as the partition 0 = to < t1 < ... < t. = T becomes dense. While a rigorous proof along these lines is difficult and involves many details, it can be given when the kernel K of R is smooth. We proceed by the easier but indirect method, via (3.2) and the simultaneous representation. The advantage of using .uw and the related measures as the catalysts stems from the properties of white noise, formally W'(t). Many have tried to calculate d,u/dv directly by getting a simultaneous representation of ,u and v in terms of independent variables. What would be needed is a set of simultaneous eigen- functions. However, these do not exist in general (note that the claim in [28], p. 334, about the existence of generalized eigenfunctions is not correct). In case v = .Mwsuch eigenfunctions do exist and are in L2 as we have seen. Basic is the fact that the covariance of white noise is formally the 6-function and 6 can be expanded in any complete set 6(s, t) = Ej oj(s)(t). Of course, this expansion is only formal, but upon integration it becomes precise. 10. Proof of Theorem 1: Sufficiency. Suppose , is a measure with mean m and cov R that satisfy (1.1)-(1.3). We will prove that ,u --. uw by actually giving d,u/duw. Let (pi, (P2, ... denote the eigenfunctions of the kernel K and define the Wiener integral

.. (10.1) = X(X) = rO (t) dX(t), j = 1, 2, Define (10.2) Fj(X) = (1 - X,)-exp [-'(Xj - k )2/(l - Xj)]/exp [-1XV2]} where kj = (k, spj), k = m'. LEMMA1. The product (10.3) F(X) = f==1 Fj(X) convergesa.e. (,uw) and is integrable iw . PROOF.First, HI (1- X,)eX3 converges because E X,2 < oo. Using EIk;< X (by (1.3)) and the 3-series theorem, it is an easy exercise to prove that

0=i (Xi - ki)2/(1 - Xi) - X -X RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 335

convergesfor a.e. X. Thus (10.3) convergesand F is a randomvariable. To prove that F e L'(,uw)we'use an idea due to Kakutani [14].Let

GN = iN=j (F.)'. We already know that GN(X) -+ (F(X))' for a.e. X. We will show that GN is a Cauchy sequencein L2(Mw). The limit of the Cauchy sequencemust also be (F)'. But L2 is complete and so (F)1 e L2(,Mw),or F e LV(Mw). To prove that GNis a Cauchy sequenceobserve that EwF, = 1, E~2(F,)' =3 2 = exp [--kI,2/(2 - Xj)](1 - Xj) /(1 - j/ Now X, , X2, are independent (MAw)and hence so are F1 , F2, * . For M < N we have by direct calculation N - (10.4) EW(GN - GM)2 = 2 - 2 Ew-(Fj) = 2(1 +i d>) Again using ? X,2 < oo and ? kj2 < oo it follows that f Pi converges.The tail of the producttends to unity and so GNis a Cauchy sequence. The lemmais proved. LEMMA2. F is the R-N derivative;F = dl/dsw . PROOF.What we must show is that (10.5) A(A) = fAF(X) dAw(X), where A is any measurableset of functions X. We prove (10,5) first for sets A of the form (10.6) A =fXj < a,, X2 < a2, **,X. < an.. Once again, X1, X2, *** are independent (MAw)and hence so are F1(X), *** Fn(X). The expectation of a product of independentvariables is the product of the expectations and by direct calculation (10.7) fAF(X) dsw(X) = Hjj7=i-'((aj - kj)/(ll - ) whereb is the standardnormal df. I Relative to ,u the randomvariable Xi is Gaussian and has mean (10.8) E,fo pj(t) dX(t) = ((pj, k) = kI. The covariance of Xl, X2, ... is (10.9) E,(Xi - k;)(Xj - kj) = f ' C'op(s)spj(t) d8dtR(s, t). Applying (1.1), we obtain (10.10) Jfo'f ,i(s)spj(t) d8dtR(s, t) = (';, (pi) - (K'pi, Xp) = (1 -As)ai. Thus X1, X2, * ** are also independent with respect to ,u. It is now easy to check that ,u(A) agrees with (10.7) and hence (10.5) follows, at least for sets of the form (10.6). In orderto prove (10.5) for any measurableset A we find by direct calculation that

(10.11) E4( 1X)j(t) - X(t))2 = 0 336 L. A. SHEPP for each fixed t. It follows that almost surely (i.w) we have

(1Q.12) X(t)= = jX),(t) and so the or-fieldgenerated by {Xi} and sets of probabilityzero (,uw)includes the measurablesets (the a-fieldgenerated by X(t), for 0 ;9 t ? T). We have alreadyproved (10.5) for the c-field generatedby {XJ} and Lemma2 follows. We have shown that when (1.1)-(1.3) hold, the R-N derivativedM/d,1w exists. This derivative is a.s. positive because the infinite product converges and it follows that ,u I-. w. Without using the positivity, the, equivalence would also follow from the dichotomy theorem of Feldman and HAjek. We turn to the other half of Theorem1. 11. Proof of Theorem1: Necessity. At this point we use an importanttheorem of Segal [21], p. 463. The followingproof of the necessity was suggestedby J. Feldman. Let ,u be a measurewith mean m and cov R and supposethat u '-, u w . Con- sider the Hilbert space H = L2[0,T] and define the bilinearform B: For soe H and V/e H define the random variables tq(qp) and I (4L) by the Wiener integral (9.9) and set

(11.1) B(so, f) = f,4 nw(4&)d,- (fn p() dM)(f n(y6) dy). The form B is positive, B(sp,sp) > 0, and is bounded, B((o,so) < (so,so) X constant (see Lemma 1 of [15]). Consequently,there is a linear transformationB [18], p. 202, with (11.2) (Bep,A) = B(qo,Aj). Since B is a positive linear transformationit has a squarerootT [181,p. 265. Segal's theorem says that if ,u -. ,uw(in his terminology nfT ..' n) we must have (11.3) T*T = B = I-K where I is the identity and K is Hilbert.Schmidt, K e V2. For all soe H, L e H we get by (11.3), (11.4) (Bqo,A) = (p, *)-(Kyp, itp)= BQp,A)*

Now choose s = 18, I = it . We have (1t) = rO dX(u) = X(t), since X(0) = 0 a.s. ,uw. We get by (11.1) (11.5) B(18,f l t) =R(s, t). By (11.4) (11.6) R(s, t) = min (,s, t) -f f0 K(u, v) du dv. This proves (1.1) necessary. RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 337

Next we prove the spectral condition on K, Xi < 1 for all j. By (11.4) we get (11.7) (p, so) _ (K4p,<) since (Bvp, p) _ 0. The largest eigenvalue of K is given by Xm. = sup,, (K<, p)/ (so, so) and so Xi < 1. To show that 1 z o(K) we may argue as follows. Suppose K(p = p, (p, v) = 1. We see that a(f) is an a.s. constant with respect to IAsince it has variance (Bp, p) = 0 by (11.4). On the other hand, with respect to lAw, t,(v) has variance one. This means I, I ,uw and we get a contradiction. This proves (1.2) necessary. (11.7) shows that if R has the form (1.1) and R is a covariance then a(K) C (- oc, 1]. Next we show that if R has the form (1.1) and sp(K) C (- oo, 1] then R is a covariance. This will prove the remark made below Theorem 1, R is nonnegative definite if and only if Xj < 1 for all j. What we must show is that (11.8) (R , ()p 0 for all soe L2. Let

f(t) = - fp(u) du. Integration by parts applied twice gives

(11.9) (R(p, p) = ib(s)>(t) d8dtR(s, t) = (1,)- (K- b,) But v(K) C (- oo, 1] and so (, b) - (Kc, b) > 0. We get (11.8) immediately. We remark that (1.3) cannot be replaced by the condition: (Ry, so) > 0 for nonzero qoe L2. Next we prove (1.3). Let v(R, m) denote the Gaussian measure with mean m and cov R sol, = v(R, m). We have

(11.10) , =u v(R, m) v (Rw, 0) = lw where Rw(s, t) = min (s, t). Ry a theorem of C. R. Rao and V. S. Varadarajan [17], p. 308, the measures must also be equivalent when the means are ignored:

(11.11) v(R, 0) r-' v(Rw, 0).

By considering the 1-1 path transformation X(t) -+ X(t) + m(t) we see that (11.11) gives

(11.12) v(R, m) v (Rw, m) = w+n Comparing (11.10) and (11.12) and using the fact that,--, is an equivalence rela- tion we get (11.13) IAw JW+M. The condition on m in order that ,uw --' lAw+mwas found by Segal [21], p. 462. It is precisely (1.3). This completes the proof of Theorem 1. In order to prove (1.4) we observe that whenever (1.1) holds K is given by 338 L. A. SHEPP

(11.14) K(s, t) = -(a9/s) (a9/t)R(s, t), s $ t. Putting (11.14) back into (1.1) we get for s < t,

(11.15) R(s, t) = s + Is (JuR12(u, v) dv) du + fJ (f R12(u, v) dv) du. Integrating on v and using R(0, t) -0 we get

(11.16) R(s, t) = s + f" Ri(u, u-) du + fo (Ri(u, t) - R1(u, u+)) du. This gives immediately

(11.17) s = f' (Ri(u, u+) Ri(u, u-)) du which is the integrated version of (1.4). 12. Proof of Theorems 2 and 3. We will now show that the product formula (10.3) for F = dM/dMwcan be expressed in terms of the Fredholm quantities. Let K denote, as usual, the kernel in (1.1) of the covariance of t -,u tw . The eigenfunctions of the resolvent H of K are (pi, .02, * * , the same as those of K, and the eigenvalues y of H satisfy yj = X1/(1 - Xj), j = 1, 2, * . . We note that E 'Ye < oo and so H has the L2 expansion (12.1) H(s, t) = EjyjVj(s)Vj(t).

With Xj = JoT(pj(t) dX(t) as in (10.1) we get formally from (12.1),

(12.2) X = J f'T H(s, t) dX(s) dX(t). Since k _ j kjp1and Hk = E kjYjyPjwe get formally

= + H(s, t)k(s) dX(t) (12.3) EjT=l X,k,/(l - Xj) BTk(t) dX(t) f T f T ds and

(12.4) Z= I kj2/(1 - Xj) = (k,k) + (Hk,k). Substituting (12.2)-(12.4) into (10.3) we get

= = [ T (12.5) F(X) H=1 Fj (X) (d(1)- exp f- - H(s,t) dX(s) dX(t)

+ B'Tk(u) dX(u) + f T Hk(t) dX(t) - -(k, k) - -(Hk, k)]. Replacing X by X + m gives (2.4). To prove (12.2) rigorously we observe that by (9.8) the partial sum

J= c (12.6) X,2'y,- Y=1yj + O rO HN(S, t) dX(s) dX(t), where

HN(S,Nt) = = N = 1, 2,

Since HN -* H in L2 we have in L2(M1W)

(12.7) fBT c fBT HN(S, t) dX(s) dX(t) -- f T c fT H(s, t) dX(s) dX(t). Now suppose that K satisfies the assumptions of Theorem 2 so that K is con- RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 339 tinuous and of trace class. It follows that the resolvent H is continuousand of trace class. H is also symmetric,of course,and it is knownthat these assumptions on H imply that (12.8) tr (H) = 7= Jo H(s, s) ds. Althoughthis fact is well knownin the theory of integral equations, apparently no publishedreference exists. Passing to the limit in (12.6) we have ' (12.9) = y + J c f H(s, t) dX(s) dX(t). Using (12.8) and ' 7cJ' H(s, t) dX(s) dX(t) = f' fJ'H(s, t) dX(s) dX(t) - f H(s, s) ds, we obtain (12.2). To prove (12.3) rigorouslywe observe that (12.10) Z7L Xkjl/(1 - Xj) = Jf'(kN(t) + HkN(t) ) dX(t), where kN = =iL kjvj . As N -> 00, kN -* k in L2. By the convergence properties of Wienerintegrals we get (12.3). Since the rigorousjustification of (12.4) is straightforwardwe have proved Theorem2. Theorem3 can be proved similarly.Instead of (12.2) we must use

(12.11) _= (Xj2 - 1)yj = f c f ' H(s, t) dX(s) dX(t), which follows from (12.6), letting N -> oo. Of course, now Ejyj does not neces- sarily converge. The trace of HK is (12.12) tr (HK) = EjyjXj and the proof of (2.5) is completedby a short calculation. We omit the proof of Theorem9 because it proceedsalong the same lines as those of Theorems1-3. 13. Proof of Theorem7. Let R be the covarianceof an n-times differentiable process Y nondegenerate at zero. Let Ao X ** * , An be defined by (5.12) and R* be definedby (5.7). We will prove that R* is a covarianceby showingthat it is the covarianceof the processZ, (13.1) Z(t) = Y(t) - Z7-noYt'f(O)Bj(t), where

(13.2) Bj(t) Z,?=oBBijRo(O,t), j = 0, * n. , and Bi, = R7j'is the inversematrix of R in (5.10). The matrixR = Rij is positive- definiteand so has a lower semidiagonal(lsd) squareroot which is unique and is given by Gauss'sformula [8], p. 37. ComparingGauss's formula with (5.13) we see that A is the lsd squareroot, that is (13.3) Rij= o AikAjk, A = Ai 340 L. A. SHEPP

i = 0, * * * , n andj = 0, * ** , n; in symbolic notation R = AAT. The covarianceof Z is by direct calculation (13.4) Rz(s, t) = R(s, t) - Z7o D=oBJkRJo(0, s)Rko(O, t). We will now show that the secondterm on the right,

(13.5) ,=o k-oB,kRjo(Oy s)Rko(O,t) = k==oAk(s)Ak(t), which will prove that R*(s, t) = Rz(s, t), the covarianceof Z. Define the matrix C= C%j,i=*= O, ,nandj= 0, nby Roo *b Ro,-1 0

(13.6) Ci, = (ai ai-l) . Rji_- 1

Rio ... Ri-1 0O wherethe last columnis zero except for the element in the jth row,j < i, which is unity. We have C(, = 0 if i < j and so C is lsd. Multiplying (13.3) on the right by R-t and adding,j = 0, n, we get (13.7) 8s = AjkR-1. ~k=uoAik I=o But using the formula(5.13) for Ajkwe get n Roo *... Rok_l Roj R7t n o (13.8) E AjkR 1 = (ak ak-1) . =Ckl. ioRko Rkk-l E RkjRzt' j-o

Putting (13.8) into (13.7) we get 5,t = Ek=oAikCk, and so C =.A-1. We see that R-1 = (AAT)-1 = CTCand so

n n n Roo ... Ro0-1 uo (13.9) ,ZRiu;Vk = E (a?za-1) j=O k=O t=0 Rio ... R1-1 Ut

Roo ... Roz-i vo

Rio ... Rttl- Vt whereuo Un , VO Vnare variables.Settingu; = Rio(Oys), j = O * , n and Vk = Rko(0, t), k = 0, * * , n we obtain (13.5). We have provedthat R* is a covariance.In fact it is the covarianceof Z. (5.8) (i) is proved. To prove (5.8) (ii) we observethat DiD2SR* is the covarianceof Z(') and by (13.1) (13.10) ZU)(0) = y(i)(0) - Z7=oY(D (O)Bj(t)(0). RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 341

But B P((0) = E O BkejRI= 5.j by (13.2) since B = R-1. Hence (5.9) holds and so (5.8) (ii) is proved. Now (5.8) (iii) follows immediatelyfrom the defi- nition (5.12) of Ao, * * , An . Note that Roo .. Roi-1 Roi (13.11) Ai() (0) = (ai ai-i). * (as/as-l)'i Rio.. * Rii-1 Rii It remainsonly to prove the uniquenessof (5.7). Suppose that (5.7) holds whereAo, ** *, A,, and R* satisfy (5.8). Differentiatingi times on s andj times ontand settings = t = Owegetfori = 0, ** ,nandj = 0, ... n, (13.12) Rij = k=o Ak Ak (O) We have used the fact that DiD2JR*(0,0) = 0 whichfollows from (i) and (ii) of (5.8). Using (5.8) (iii) we see that A = Aki)(0) is an lsd squareroot of R which we know to be unique. Now we differentiate(5.7) j times on s only and then set s = 0. Again, DlIR*(0, t) = 0 and we get

(13.13) Rjo(O0t) = Zs"'=oA ) (O)Ai(t)X j = 0, *. , n. The matrix A = Ai("(O) is nonsingularbecause of the nondegeneracyat zero and so the linear equations (13.13) have a uniquesolution Ao(t), * , An(t) for each fixed t. We have proved Theorem7. 14. Proof of Theorem 8. Let ,u be a measurewith mean m and cov R. The proof of Theorem8 will be based on the decomposition(5.7). Given a process Z with covarianceR* we may realize a process Y with covarianceR by setting

(14.1) Y(t) = E7.ont,Aj(t) + Z(t) where to, ** *, in are standardnormal variates independentof Z. Using (14.1) we will obtain the simultaneousrepresentation of ,uand ,Aw.. Supposethat m and R satisfy the hypothesisof Theorem8. Let , , ..2 be the complete o.n. system of eigenfunctionsof K in (5.14) and set (14.2) 4fn-(t) = Jf'[(t - u)'/n!]Vpj(u)du. The processW. has the representation (14.3) Wn(t) = 7=ot'/j! + n(t) where to, ** *, XX1), are independentstandard normal variates on some space. The process Y definedon the same space as the i's and q's by (14.4) Y(t) = ot=oj,Aj(t) + I=. [kI + (1 - Xj)'17jI'ij,n(t)+ f=Om(0)t'/j! has mean EY(t) = =1k=4,n (t) + E7= m(j)(O)tl/j! = m(t) 342 L. A. SHEPP

by (5.6) and covariance R in (5.7) as a short calculation shows. Now it is easy to obtain a formal expression for dMI/dMw,. We compute the formal likelihood ratio under the two measures as follows: We expand (formally) (14.5) X(t) = =oX()(O)titi!X + ? dX1Xt(u) X, = fTpj(u) and denote Xi = X(i)(0). Under ,'wn we have Xi = (i, 2 = t7j. Under ,u we get with mt = mt(0) from (14.4), (14.6) Xi = mi + Z7=o t A ()(O), (14.7) Xj = kj + ,j(l - Xj)' + ELo {iaijX where aij = (ai, (pj). Inverting (14.6) with C = A-1 given by (13.7), we get (14.8) tj = ej= =oCi(X' - mi). Therefore,

(14.9) i= (Xj - kj - uj)/(l- , Uj ==E=oeiaij. The likelihood ratio becomes (14.10) dM(X)/dAwn(X)

- - X = exp [-2 Z=o (ej2- (Xj)2) o (,2 2)]IJI where IJIis the Jacobian of the transformation (14.6), (14.7) of X', Xj tj, ,j* Now jatja/XIj = lCjil = Coo... Cnnbecause C is lsd. Now Cii = 1/Aii and Aii = (ai/ai-i)' by (5.13). We get Coo... Cnn= (an) . We have laii/aXjI = [][Ij (1 - Xj))]- = [d(l)]-' and so IJI= (d(l)an)-. Now, I (nq2_ X 2) = ?=1 [(X,-kj)2/(1 -Xj) - X] -2 , (Xj - kj)ujl(1- X) + E UI2/(1 Xj). Forthe firstterm on the rightsee (12.2)-(12.4). Forthe otherswe obtain (14.11) E=i1 (Xj - kj)uj = oei fo'dX0f(t),a1(t)

,ji=i (Xj - kj)ujXj/(1 - Xi) = E n=oe0 fbo fbo H(s, t)ai(s) ds diZ8) (t), whereX(t) = X(t) -m(t) and also

(14.12) =E k=Oeiek(ai, ak), 1 U j2Xj/(1-X1) = Z=o EkZ==Oeiek(Hai , ak). Now (14.11) and (14.12) give (5.16). The formalcalculations can be made precisewhen the hypothesis-(5.6), (5.14) and (5.15)-of Theorem8 holds. Thisprocedure is an imitationof the sufficiencyproof for Theorems 2 and3 andis omitted. The necessityof (5.14) is easy becauseif X is a processwith X(t) Wn(t), RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 343 it follows from Theorem 6 that Xf)(t) - X( () W(t), and so (5.14) follows from (1.1). To prove the necessity of (5.6), suppose X is a process with mean m and X -Wn . It follows, see Section 12, that W. -' Wn + m. Using the transforma- y(n) tion Y(t) (t) - y(n) (0) on both sides of the latter equivalence we obtain (14-13) Wn(n (t) _ Wn() )(O) , W,(n) (t) _ W,(nf)() + m(r)() - m(nf)() But W(t) = Wn(n)(t) - W.(n)(O) and so W ^-' W + m(n1 (t) - m(n)(O). Now (1.1) gives (5.6) immediately. The necessity of (5.15) is formally obvious but apparently is difficult to prove. We proceed as follows: Suppose X is a process and X Wn . Using the simul- taneous representation of X and Wn, (14.3) and (14.4), we see that the sequences of random variables S31 = t, j = 1, 2, ***, and Sj2 = (1 - 'X)'ni + kj + = aijq, j = 1, 2, * , are equivalent. Here to, * * *, n i, ... are inde- pendent, standard normal variates. In order to prove (5.15) we must show that (14.14) i , i =O, 1, * ,n.

Let I,uand 4t2be the measures induced by S' and s2 respectively on the space of infinite sequences. Then I,u ,U2 and so the Hellinger integral [14]

(14.15) f (dAj d,2)1> 0.

The integral can be evaluated explicitly. Given to, * * n,i both S' and S2 are sequences of independent random variables and so

= - + (14.16) f (d,l dM2)' r- (-2(o2 + tn ... 1j=1 rj(6x *** an)dto d. n where (14.17) r1 (27r) f-'o exp (-4x2) exp (-4(x - b )2/(1- Xi)) dX/(j and bj = Ik + $~'=oaijti, j = 1, 2, *-- . Evaluating the integral (14.17), (14.16) becomes

exp + + (14.18) A*r- X (-(& 2 ** tn )) fl7=,exp (-b12/4(2 - X)) do ... dtnX where (14.19) A = = (1 - Xi)/(1 - Xj/2)'. The product (14.19) convergesbecause Z Xj < oo. Now (14.18) is positive by (14.15) and it follows that (14.20) 0=0IV < X for a set of so%,* *, in of positive measure in Euclidean n + 1 space. We have already proved the necessity of (5.6) and so E kj2 < oo. Thus 2 (14.21) = < 00 344 L. A. SHEPP

for a set of o, *... , in of positive measure. It follows readily that (14.14) must hold and so (5.15) is proved.This completesthe proof of Theorem8. IV. EXAMPLES OF R-N DERIVATIVES The examplesgiven here will be appliedin the next section in orderto obtain new informationabout W and other processes.They will also serve to illustrate how Theorems2 and 8 are applied. 15. Scale changes of W. Let h be an absolutely continuous and increasing function on [0, T] with h(O) = 0. Define the process (15.1) Z(t) = [h'(t)]-'W(h(t)), 0 < t < T. As can be seen, the differentialincrements of Z have the same varianceas those of W. Undervery generalconditions on h, which are given preciselyby the follow- ing theorem, Z W. THEOREM11. Z W if and onlyif h' = 92 whereg is absolutelycontinuous and 9 =^yeL. For smooth h, say h e C3 the R-N derivative.is (15.2) (dyz1/dyw)(X) = (h'(T)lh'(0))' *exp {-[X2(T)/4]h"(T)/h'(T) f I f(t)X2(t) dt} wheref = -2((h"/h')' - 1(h"/h')2), theSchwarzian derivative of h. Doob [5] consideredthe more general scale changeZ(t) = v(t)W(u(t)/v(t)) with covariance (15.3) R(s, t) = u(min (s, t))v(max (s, t)). Such covarianceswere called triangularby Varberg[25], who calculatedthe R-N derivativefor equivalenttriangular processes by evaluating directly the limit of the finite dimensionaldensities. The densitiescan be written explicitlybecause Z is a Markov process. In orderto prove the theoremwe find from (15.3) that (15.4) K(s, t) =-u'(s)v'(t), s < t, = -u (t)v'(s), t < s, where u = h/(h')', v - (h')-i. Now if y e L then v' = y, u' = hy + jgj-1, are both in L2 and so K e L2.Conversely if K e L2then v' and hence y e L2. We will see later that 1 t a (K) is automaticallytrue and so by Theorem1 we have proved that y e L2is necessaryand sufficient. Now assume that h is smooth, say h e C3. We make this assumptionin order to use integrationby parts at one step to expressd,AzId,Aw in as simple a form as possible. We will calculatethe Fredholmdeterminant and resolventof K at A = 1. The eigenvalueproblem K(p = Xp can be written

(15.5) -v ' (u'qs) - u' t (V ) = X

(15.8) f -2[(h"/h')'-2 and so R is in fact a Green function. Letting y(t) = so, a direct calculation shows that the eigenvaluesX of (15.5) are those Xfor which (15.9) y" = (1 - X71)fy, y(0) = 0, y'(T)/y(T) = (1 - ?J')v'(T)/v(T) has a nontrivialsolution y. Now it is clearthat X = 1 e sp (K) becauseif K4o= s then by (15.9) y = 0 and so s = 0. To handle nonsmooth h, we would replace (15.6) and (15.9) by integrated versions, integral equations. The details are straightforward. We want d(1) = II (1 - X) where X, satisfy (15.9). Define for any complex ,Bthe uniquesolution yp to (15.10) Y"= ffY#, y(0) = 0, yO.'(0) = 1. Set

(15.11) D(A3) = yp'(T) - flyp(T)v'(T)/v(T). It is known from the general theory of differentialequations that yp(T) and yp'(T) depend analytically on 13.Furthermore it is not difficult to show that yp(T) is entire and of order 1. Also yp'(T) is entire and of order 1. It follows that D(Al) is entire and of order 2-hence D(13) is its own canonicalproduct: (15.12) D(13) = II (1 - 0/13>), where f,1 ?32, *... are the zeros of D. Now, we have (15.13) 13 =1 _ -1 becauseD(13) = 0 if and only 13= 1 - X71where X satisfies (15.9) for some non- zero y = yp (note that y,Bis nonzerobecause yp'(0) = 1). Therefore d(l) = II(1-X,) = (II (1- -7))1 = (D(1)V'. What is D(1)? By (15.11), (15.14) D(1) = [y1'(T)v(T) - y1(T)v'(T)]/v(T). For 13= 1 (and only for this value),

(15.15) yA'(t)v(t) -yp(t)v'(t) = constant being the Wronskian. Putting t = 0 to evaluate the constant we get (15.16) D(1) = v(0)/v(T) = (h'(T)/h'(0)j). 346 L. A. SHEPP

Finally, (15.17) d(l) = (h'(O)/h'(T))'. Next we calculate the resolvent kernel H. It will turn out that (15.18) H(s, t) = 0(max (s, t)) where (15.19) @'(t)= -f(t), @(T) = lh"(T)/h'(T) H is the unique continuous solution to (15.20) H-K= HK =KH. It is straightforward to check that (15.18) satisfies (15.20) and so (15.18) is proved since H is unique in (15.20). By (2.4) we have since m = 0, (15.21) (d,uz/d,uw)(X)= [d(1)] iexp [-1 fJTfJT H(s, t) dX(s) dX(t). Applying (9.13) or proceeding formally (note that X(0) = 0) we have

Co Co H(s, t) dX(s) dX(t) fT (15.22) = 2 (f'H(s, t) dX(s)) dX(t) fT = 2 (ft0 (t) dX(s)) dX(t) = fT 0(t) dX2(t) = X2(T)O(T) + fjX2(u)f(u) du. We obtain (15.2) immediately and Theorem 11 is proved. Let W* be the pinned Wiener process, (15.23) W*(t) = W(t) - tW(l), 0 < t < 1. The scale change (15.24) Z(t) = [h'(t)]-IW*(h(t)) will have Z W* when h satisfies the hypotheses of Theorem 11 and in addition (15.25) h(l) = 1. By identical techniques, this time using Theorem 9, one shows that the R-N derivative is (15.26) (d,.z1/dyw*)(X) = (h'(O)h'(1))*i exp [-- f f(t)X2(t) dt] where f is again the Schwarzian derivative of h. (15.26) will be applied in Section 18 in order to evaluate certain integrals. 16. The linear covariance. Consider the measure , with m = 0 and covariance (16.1) R(s, t) = '(1 - It - sl) RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 347

with T < 2. Now ,u',-'w and we will use the decomposition method of Theorems 7 and 8 in order to obtain dMu/dMuw0. Recall that Wo(t) = to + W(t). For n = 0 and m = 0, (5.16) becomes (dM/dMwo)(X) = [d(1) R(O, 0)f-? exp [- JoJoTfT H(s, t) dX (s) dX (t) (16.2) + 2X2(O) ( -[R(O, O)]-1((I + H)ao, ao))

+ X(O)[R(O, 0)]f fof (I + H)ao(t) dX(t)]. In the case (16.1) we have Ao(t) = (1 - t)/2", ao(t) =-1/21. We get in turn R*(s,t) =s- st/2; s< t,

(16.3) K(s, t) =2,

H(s, t) = 1/(2 -T), so that K and H are constants. The spectrum of K is the single element X = T/2. We must exclude T 2 by (1.2) and indeed for T = 2 it is clear that A I Awo since Z(2) = -Z(O) a.s. for A = Az . We obtain d (1) = 1- T/2. We obtain further

(I + H)ao(t) = ao(t) + foT H(t, s)ao(s) ds (16.4) = -22/(2 -T) ((I + H)ao, ao) = T/(2- T), g 'fg H(s, t) dX(s) dX(t) = (X(t)-X(O))2/(2 - T). We put all the above ingredients into (16.2) and we get (16.5) (dA/dAwo)(X) = [2/(2 - T)'] exp [X2(0)/2 - (X(O) + X(T) )2/2(2 - T)]. (16.5) can also be obtained by a direct passage to the limit from the finite dimensional densities of ,u which have been found explicitly by Slepian [23]. The results agree. We draw attention to the normalization inherent in considering the Wiener process with unit diffusion constant. A process Z may be equivalent to oW, for some a- # 1, rather than to W itself. This is only a scale factor and gives no trouble. We get Z/l- W and (16.6) (dAz/dAaw)(X) = (dAz,o/dAw)(Qh-'X). Sometimes o- = 1 is not the most convenient normalization. Instead of (16.1) it is more usual to consider (16.7) k(s, t) = 1 -It - SI. Let us denote by ,uthe measure with cov R and mean 0. Let juodenote the measure ,u conditioned so that X(0) = xo given. The mean of guois

(16.8) fho(t) = xoR(0 t)/R(0, 0) 348 L. A. SHEPP

and the coy of -po is (16.9) Ro(s, t) = f(s, t) - R(0, s)R(0, t)/R(0, 0). go is a Gaussian measure and is equivalent to the Wiener-typemeasure cor- responding to the process (16.10) w(t) = x0 + 21W(t), 0 < t < T. The R-N derivative is easy to obtain and is another form of (16.5); it can be computed also from (2.4). We have (16.11) (djo/dAto)(X) =[2/(2 - T)]' exp (2x02) exp [-2(Xo + X(T) )2/2(2 - T)]. We will apply (16.11) in Section 17 to solve a certain first passage problem. The exampleof this section was chosen for its simplicity. We can obtain with our methods all known examples of R-N derivatives in the Gaussian case as well as many new examples. The calculations, although straightforward,are sometimes- quite tedious. We have given illustrationsby exampleof all the main theorems.The reader may find it instructiveto obtain the R-N derivativein the Ornstein-Uhlenbeck case (Example3 of [251)by our methods. V. SOME APPLICATIONS OF R-N DERIVATIVES The R-N derivative has found its main applicationin statistics as the likeli- hood ratio. However, it can also be used as a theoretical tool, as Skorokhod [22], p. 408, pointed out. We will use it both to calculate probabilitiesand to evaluate function space integrals. 17. Calculatingprobabilities. We shall prove that if b > 0, (17.1) Pr {W(t) < at + b,0 < t < T} = 1((aT + b)/T') - eC2((aT - b)/T'). We see that (17.1) is the probabilitythat a gamblerwith income does not go brokein time T < oo if b representshis initial capital, a his fixed rate of income (or outgo), and W his losses due to chance fluctuations. (17.1) was found by Doob [5] for T = oo and was given in generalby Malmquistin a differentform and by a differentmethod [16]. To prove (17.1) we observe that by definitionof dM/cd4w,

(17.2) #i'(A) = fA (dt/d/w)(X)dAw for any event A. Let ,u = Mw+mwhere m(t) = -at and take A = {X: X(t) < b, 0 < t < T}. By (2.4) we obtain since k(t) = -a,

(17.3) F(X) = (d,ud,uw) (X) = eaX(T)e-Aa2T RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 349

Now by (17.2),

(17.4) A(A) = eaf A eaX(T)dgw(X) = ea2Tf ebZa Pr {M(T) < b; W(T) = x} dx where M(T) = maxo?e?T W(t). But the reflection principle gives for x < b, (17.5) Pr {M(T) < b; W(T) = x} = Pr {W(T) = x} - Pr {W(T) = 2b - x}. Using

Pr {W(T) = x} = (27rT)-IeIZ21T and ,u(A) = Pr {W(t) < at + b, 0 < t < T} we obtain (17.1) after a simple calculation. The point of using (17.2) is that the reflectionprinciple fails for slanted lines at + b, while (17.2) enables us to reduce the problemto the hori- zontal line. As a second exampleof the same technique,we solve a first passage problem originallygiven by Slepian [23] in a differentform. Let S = S(t), 0 < t < T be the process with mean zero and covariance

(17.6) R(s, t) = 1-ft-sf, It-sI < 1, = 0, It- sI > 1. Suppose T < 1. Let (17.7) Pa+(T I xo) = Pr {S(t) < a, 0 < t < T S(O)= xo}, xo < a, Paj(T I xo) = Pr {S(t) > a, 0 < t < T I S(O) = xo}, xo > a, be the first passageprobabilities. We shall prove

(17.8) PaG-(TI xO) = 4( 4 [a- xo( - T)]/[T(2- T)]- )

-e-(a2X02)(_(_i[xo - a(1 - T)]/[T(2 -T)] ) The formulas break down for T > 1 and in this case the problem remains tun- solved. Let So denote the processS conditionedto pass throughxO at t = 0. We have seen that So ,-- w = xo + 21W and the R-N derivative is given by (16.11). Let A = {X: X(t) < a,0 9 t < T}.

Applying (17.2) with ,u = ,tso and Mw replaced by ,u. we get (17.9) PaJ(T Ixo) [2/(2 - T)]yeX02JfAexp [-l(xo + X(T))'/2(2 - T)]dgw(X). Using the reflection principle to evaluate the integral, we obtain (17.8). P- is 350 L. A. SHEPP

obtained in a similar way. In comparing our form of the answer with Slepian's note that his Qa(T Ixo) is the derivative on T of our Pa(T Ixo). In principle one could use the method of this section to solve other first passage problems. However, these are the only cases in which we could evaluate the integrals explicitly. 18. Evaluating some special integrals. The integral of the R-N derivative over all space is unity. This fact will permit us to evaluate the Wiener integral of any functional that is the exponential of a quadratic form. Some special cases of this result are due to Kac and Siegert [12] who obtained them by other methods. Their results were later simplified by Anderson and Darling [1]. We will then apply the evaluations to prove a dichotomy theorem about Wiener paths. We begin with a general identity. Let L be any symmetric continuous kernel of trace class. Then

(18.1) Ew exp ['Xfr rf L(s, t) dX(s) dX(t)] = (DL(X))-2f whereEw is the Wiener integral, dL is the Fredholm determinant of L, and X satisfies (18.2) 1-XXj>O, j=1,2, 2 . where X1, 2, * ** are the eigenvalues of L. When (18.2) fails the intearal is + oo. To prove (18.1) note that by (2.4) we have (18.3) 1 = Ew[d(1)]-2 exp [-fr f' H(s, t) dX(s) dX(t)] where d(1) = dK(1). Now if y denotes eigenvalues of H theny - X = YXand so

(18.4) dK(1) = llA (1 - X) = II (1 + y)-1 = d (_1) = d-H(1). Let now L = -XH and we obtain (18.1). Wlhen (18.2) fails the quadratic form

(18.5) ( s-X(Lso,P ) is not positive-definite and it is at least formally clear that (18.1) should diverge. The proof is omitted. It is amusing to expand both sides of (18.1) in powers of X and check coefficients of XA.The general equality seems to be difficult to prove directly. Next we evaluate (18.6) A (f) = Ew exp [- Ifr f(t)X2(t) dt] for any, say continuous, f. Solve the one-point problem (18.7) 9"-fg, g'(T) = 0, g > 0 on [0, T). Then

(18.8) A(f) = (g(T)/g(O))2. When (18.7) fails to have a solution positive on the half-open interval [0, T) then RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 351

A (F) = + co. The result is also valid for T = co provided lim g(x) = g((o ) > 0 and lim g'(x) = g'( co) = 0. Suppose now that the basic process is W*, the pinned Wiener process, instead of W. We have

(18.9) Ew exp [-2 ff(t)X2(t) dt] = (l(1))Y where 1 is the unique solution to (18.10) 1" = fi, 1(0) = 0, l'(0) = 1, provided 1 > 0 in (0, 1]. When the solution 1 of (18.10) has a zero in (0, 1] then (18.9) is +oo. For positive f, (18.9) is finite and was obtained by Anderson and Darling [1] who simplified a previous formula due to Kac and Siegert [13]. To prove (18.9) we proceed as follows. By (15.26) we have for any h with h(0) = 0, h(l) = 1 and h' > 0,

(18.11) Ew exp [-2J0 J(t)X2(t) dt] = (h'(0)h'(1))* wheref= - ((h"/h')' - 2(h"/h')2). Now choose h to satisfy

(18.12) h(0) = 0, h'== o - where g is any positive solution of (18.13) 9 = fg. We have h > 0, h(1) = 1 and the Schwarzian of h is f, (18.14) f = - ((h"/h')' - (h"/h')2) = 9 /= f. By (18.11) we have

(18.15) Ew exp [-_Jf f(t)X2(t) dt] = (g(0)g(j)f1 g-2)-1. Let (18.16) 1(t) = g(0)g(t)fo g-2 It is easy to check that 1 satisfies (18.10) and this gives (18.9). Now it is known from Sturm-Liouville theory that a positive solution g exists to (18.13) if and only if the solution 1 of (18.10) is positive in (0, 1]. It is easy to see that when 1 has a zero then (18.9) is actually + oo.A similar proof can be given for (18.8). The formula (18.9) for W* is more symmetric than (18.8) for W as is evident by (18.15). This is because W* is time-reversible. We have assumed f to be continuous but this is not necessary. In general, one expresses the differential equation g" = fg, g'(T) = 0 in the form of an integral equation (18.17) g(t) = g(T) + fT (u - t)g(u)f(u) du. One can now allow f to be simply measurable. Further, f can be replaced by a measure: fdu = dm. 352 L. A. SHEPP

As mentioned above A (f) has been calculated for continuous f > 0 by Kac and others by using various techniques. However, the most direct way to obtain A (f) in this case was found by Gel'fand and Yaglom [29]: By Riemann integration and bounded convergence we have

(18.18) A(f) = limn-oEw exp [-4Znf(jT/n)X2(jT/n)T/n]. Now the expected value on the right is an n-dimensional integral that can be evaluated explicitly. Letfj = f(jT/n), x; = X(jT/n). The integral in (18.18) is

(18.19) .fr . . exp [-2 >fiX T/n - (Xj-x11)2n/T] dxl ... dxn/ (27rT/n) = (Ib.7 |1 ) where b, r=) 2 + fjT/n2, i = j =1, , n-1, ji - jl=1, =1 + fnT2/n2, i = j =n, = ), 1i- Al > l Define n) ... (18.20) gk =gk jbwt in, k =1, 2, n.

We see that gn = 1 + fnT2/n2, gn n-1 = O(T2/n2) and

(18.21) A29k = fk+lgk, k = 1, * * , n - 1. We compare (18.21) with the differential equation

(18.22) 9" = fg, g(T) = 1, g'(T) = 0, with g(kT/n) gk . Standard techniques of comparison of difference and dif- ferential equations give

(18.23) A(f) = lim (g,(fn) = (g() which agrees with (18.8) since g(T) = 1. If?, and v are Gaussian then dil/dv(X) is always the exponential of a quadratic. form in X. With v = ,uw, the quadraticform is diagonal, (18.24) fJ X2(u)f(u) du, if and only if ,- = uz where Z- is a scale change as in (15.1). Indeed, as we have seen, (18.24) holds for j, = ,Iz . Conversely, we can obtain (18.24) for any f by means of a properly chosen scale change. Thus (18.24) characterizes scale change processes. 19. A dichotomy theorem for Wiener paths. Suppose f > 0 on [0, T]. The integral (19.1) f f(u) W2(u) du RADON-NIKODYM DERIVATIVES OF GAUSSIAN MEASURES 353 either convergesa.s. or diverges a.s. according as (19.2) f 'uf(u) du convergesor diverges. The convergence is trivial because if (19.2) is finite then ' (19.3) Ew(f f(u) W2(u) du) = fo uf(u) du < 0o and so (19.1) must be finite a.s. Now suppose fr uf(u) du = oo. We will prove that

(19.4) Ew exp [-f Tf(u)X2(u ) du] - 0 and so fo f(u) W2(u) = oo a.s. will follow. In order to avoid details we assume that f is continuous away from zero. By (18.8) it is enough to prove the follow- ing simple lemma. LEMMA.Suppose f is continuous on (0, T] and fr" uf(u) du = oo. Then (19.5) 9" = fg, g'(T) = 0, g(T) = 1, has g(0) = oo . PROOF. Since g is convex and g'(T) = 0, g is monotonically decreasing. Sup- pose g(0) = M < oo. Then 0 < -Eg (e) g9(0) - g(e) < M by convexity. We have, since g > 1, for any e > 0 (19.6) fe uf(u) du < f "uf(u)g(u) du = Je ug (u) du = Tg (T) - Eg(e) - g(T) + g(E) < 2M. This contradicts fr uf(u) du = oo and proves the lemma. AN EXAMPL.E.Let f(u) = 2/(e + u)2, 0 < c, T < oo. The solution to (18.7) is

(19.7) g(u) = A(e + u)-1 + B(e + U)2, A = 2B(e + T)3. We obtain

(19.8) Ewexp [-f' (E + u)-2W2(U) du] = (3e/2(e + T)(1 + (e/e + T)3/2))'. For other examples where (18.7) has an explicit solution see [1]. Letting e -* 0 we have by monotone convergence

(19.9) Ew exp [-f 0u-2Wf2(Wu) du] = 0 and so

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